Supplement: Risk factors for mortality among hospitalized patients with COVID-19

1 Dependencies and settings

# R packages
library("corrr") # 0.4.2
library("cowplot") # 1.0.0
library("dplyr") # 1.0.0
library("DT") #0.14
library("ggplot2") # 3.3.1
library("oem") # 2.0.10
library("gridExtra") # 2.3
library("knitr") # 1.28
library("kableExtra") # 1.1.0
library("magrittr") # 1.5
library("mice") # 3.9.0
library("pROC") # 1.16.2
library("purrr") # 0.3.4
library("rcompanion") # 2.3.25
library("rms") # 6.0.1
library("splines") # base
library("tableone") # 12.0.0
library("tibble") # 3.0.1
library("tidyr") # 1.1.0

# Namespace clash
select <- dplyr::select

# Settings
set.seed(123)
theme_set(theme_bw())
n_imputations <- 2    # Number of multiple imputations with MICE
n_boot_val <- 40      # Number of bootstraps with rms validation and calibration
n_boot_probs <- 100   # Number of bootstraps for computing predicted probabilitiy CIs
n_rep <- 2            # Number of repeats with grouped lasso
CACHE <- FALSE         # Whether to use cached results for long computations

2 Data

We begin by loading the “training” dataset and restrict to (i) patients age 18 and older and (ii) with an index date more than 2 weeks prior to the data release. Note that we are not unfortunately allowed to publicly share this data.

2.1 Load

filter_ie <- function(data){
  data1 <- data %>%
    filter(age >= 18) 
  data2 <- data1 %>%
    filter(as.Date("2020-06-05") - index_date > 14)
  n_dropped <- nrow(data1) - nrow(data2)
  percent_dropped <- formatC(100 * n_dropped/nrow(data1), 
                             format = "f", digits = 2)
  message(n_dropped, " (", percent_dropped, "%)",
          " patients were dropped due to the 2-week cutoff.")
  return(data2)
}

train_data <- readRDS("train_data.rds")  %>%
  filter_ie()

## 750 (5.21%) patients were dropped due to the 2-week cutoff.

2.2 Clean

# Recoding
train_data <- train_data %>% 
  mutate(
    ## "died" in add_deaths() should be an integer
    died = as.integer(died),
    
    ## Convert unknown or other/unknown to missing
    race = ifelse(race == "Other/Unknown", NA, race),
    ethnicity = ifelse(ethnicity == "Unknown", NA, ethnicity),
    sex = ifelse(sex == "Unknown", NA, sex),
    
    ## Oxygen saturation should have plausible values
    spo2 = ifelse(spo2 == 0, NA_real_, spo2),
    spo2 = ifelse(spo2 > 100, 100, spo2)
  ) %>%
  
  ## Better names for some variables
  rename(diabunc = diab, cci = score) %>%
  
  ## CCI should be an integer
  mutate(cci = as.integer(cci)) %>%
  
  ## Convert omorbidities to character
  mutate_at(
    c("ami", "chf", "pvd", "cevd", "dementia", "cpd", "rheumd", "pud",
      "mld", "diabunc", "diabwc", "hp", "rend", "canc", "msld", "metacanc",
      "aids", "hypc", "hypunc"),
    function (x) ifelse(x == 1, "Yes", "No")
  ) 

# Create "derived" variables

## Function to add race/ethnicity variable to dataset (done after separately 
## imputing missing race + ethnicity information)
add_race_ethnicity <- function(data){
  data %>% mutate(
    race_ethnicity = case_when(
      race == "Caucasian" & ethnicity != "Hispanic" ~ "Non-Hispanic white",
      race == "African American" & ethnicity != "Hispanic" ~ "Non-Hispanic black",
      race == "Asian" & ethnicity != "Hispanic" ~ "Asian",
      is.na(race) | is.na(ethnicity) ~ NA_character_,
      TRUE ~ "Hispanic"
    ),
    race_ethnicity = relevel(factor(race_ethnicity), ref = "Non-Hispanic white")
  )
}

train_data <- train_data %>%
  mutate(
    calendar_time = as.numeric(index_date - min(index_date)),
    index_month = as.integer(format(index_date, "%m")),
    death_month = as.integer(format(date_of_death,"%m")),
    os_days = pmax(0, date_of_death - index_date),
    
    ## Categorize BMI
    bmi_cat = case_when(
      bmi < 18.5 ~ "Underweight",
      bmi >= 18.5 & bmi < 25 ~ "Normal",
      bmi >= 25 & bmi < 30 ~ "Overweight",
      bmi >= 30 ~ "Obese",
      TRUE ~ NA_character_
    ),
    
    ## Combine comorbidities
    diab = case_when(
      diabunc == "Yes" | diabwc == "Yes" ~ "Yes",
      TRUE ~ "No"
    ),
    hyp = case_when(
      hypc == "Yes" | hypunc == "Yes" ~ "Yes",
      TRUE ~ "No"
    )
  ) %>%
  
  ## Create race/ethnicity variable
  ## Later to be created after imputation from imputed race + ethnicity
  add_race_ethnicity() 

2.3 Label

# Labels
## Categorical variables
demographic_cat_vars <- tribble(
  ~var, ~varlab,
  "sex", "Sex",
  "race", "Race",
  "ethnicity", "Ethnicity",
  "division", "Geographic division",
  "smoke", "Smoking",
  "race_ethnicity", 'Race/Ethnicity'
) %>% 
  mutate(group = "Demographics")

comorbidity_cat_vars <- tribble(
  ~var, ~varlab,
  "ami", "Acute myocardial infarction",
  "chf", "Congestive heart failure",
  "pvd", "Peripheral vascular disease",
  "cevd", "Cerebrovascular disease",
  "dementia", "Dementia",
  "cpd", "Chronic pulmonary disease",
  "rheumd", "Rheumatoid disease",
  "pud", "Peptic ulcer disease",
  "mld", "Mild liver disease",
  "diabunc", "Diabetes (no complications)",
  "diabwc", "Diabetes (complications)",
  "hp", "Hemiplegia or paraplegia",
  "rend", "Renal disease",
  "canc", "Cancer",
  "msld", "Moderate/severe liver disease",
  "metacanc", "Metastatic cancer",
  "aids", "AIDS/HIV",
  "hypunc", "Hypertension (no complications)",
  "hypc", "Hypertension (complications)",
  "diab", "Diabetes", # Combines diabunc and diabwc
  "hyp", "Hypertension", # Combine hypunc and hypc
) %>%
  mutate(group = "Comorbidities")

vital_cat_vars <- tribble(
  ~var, ~varlab,
  "bmi_cat", "Body Mass Index (BMI)",
) %>%
  mutate(group = "Vitals")

cat_vars <- bind_rows(demographic_cat_vars,
                      comorbidity_cat_vars,
                      vital_cat_vars)

## Continuous variables
demographic_continuous_vars <- tribble(
  ~var, ~varlab,
  "age", "Age",
  "calendar_time", "Calendar time"
) %>%
  mutate(group = "Demographics")

comorbidity_continuous_vars <- tribble(
  ~var, ~varlab,
  "cci", "CCI",
) %>%
  mutate(group = "Comorbidities")

vital_continuous_vars <- tribble(
  ~var, ~varlab,
  "bmi", "Body Mass Index (BMI)",
  "dbp", "Diastolic blood pressure",
  "sbp", "Systolic blood pressure",
  "hr", "Heart rate",
  "resp", "Respiration rate",
  "temp", "Temperature",
) %>%
  mutate(group = "Vitals")

lab_vars <- tribble(
  ~var, ~varlab,
  "alt", "Alanine aminotransferase (ALT)",
  "ast", "Aspartate aminotransferase (AST)",
  "crp", "C-reactive protein (CRP)",
  "creatinine", "Creatinine",
  "ferritin", "Ferritin",
  "d_dimer", "Fibrin D-Dimer",
  "ldh", "Lactate dehydrogenase (LDH)",
  "lymphocyte", "Lymphocyte count",
  "neutrophil", "Neutrophil count",
  "spo2", "Oxygen saturation",
  "pct", "Procalcitonin",
  "tni", "Troponin I",
  "plt", "Platelet count (PLT)",
  "wbc", "White blood cell count (WBC)"
) %>%
  mutate(group = "Labs")

continuous_vars <- bind_rows(
  demographic_continuous_vars,
  comorbidity_continuous_vars,
  vital_continuous_vars,
  lab_vars
)

# All variables
vars <- bind_rows(cat_vars,
                  continuous_vars)

get_var_labs <- function(v){
  vars$varlab[match(v, vars$var)]
}

3 Preliminary data analysis

Before starting modeling, it’s a good idea to carefully inspect the data.

3.1 Sample size

train_data %>%
  count(name = "Sample size") %>% 
  mutate(Data = "Optum training set") %>%
  select(Data, `Sample size`) %>%
  kable() %>%
  kable_styling()

Data	Sample size
Optum training set	13658

3.2 Missing data

missing_df <- train_data %>%
  select(one_of(vars$var)) %>%
  mutate_all(function (x) ifelse(is.na(x), 1, 0)) %>%
  mutate(id = factor(1:n())) %>%
  pivot_longer(cols = vars$var, names_to = "var", values_to = "missing") %>%
  left_join(vars, by = "var")

3.2.1 Proportion missing

# Compute proportion missing
prop_missing <- missing_df %>%
  group_by(varlab) %>%
  summarise(prop = mean(missing))

## `summarise()` ungrouping output (override with `.groups` argument)

# Plot
ggplot(prop_missing, aes(x = varlab, y = prop)) +
  geom_bar(stat = "identity") + 
  geom_text(aes(label = formatC(prop, format = "f", digits = 2)),
          nudge_y = .03, size = 3) +
ylim(c(0, 1)) +
xlab("") +
ylab("Proportion") +
coord_flip() +
scale_x_discrete(limits = rev(sort(vars$varlab)))

3.2.2 Missing by patient

ggplot(missing_df,
       aes(x = id, y = varlab,  fill = factor(missing))) +
  geom_raster() + 
  theme(axis.title.x = element_blank(),
        axis.text.x = element_blank(),
        axis.ticks.x = element_blank(),
        axis.title.y = element_blank(),
        legend.position = "bottom") +
  scale_fill_manual(name = "Missing",
                    values = c("lightgrey", "steelblue"),
                    labels = c("No", "Yes")) +
  scale_y_discrete(limits = rev(sort(vars$varlab)))

## Warning: Raster pixels are placed at uneven vertical intervals and will be shifted. Consider using geom_tile() instead.

missing_df %>% 
  # Count of missing by patient
  group_by(id) %>%
  summarise(n_missing = sum(missing),
            prop_missing = n_missing/n()) %>%
  
  # Plot
 ggplot(aes(x = prop_missing)) +
  geom_histogram(binwidth = .03, color = "white") +
  scale_x_continuous(breaks = seq(0, 1, .05)) +
  scale_y_continuous(n.breaks = 20) +
  xlab("Proportion of covariates that are missing") +
  ylab("Count") 

## `summarise()` ungrouping output (override with `.groups` argument)

3.3 Distributions

3.3.1 Covariates

3.3.1.1 Categorical

cat_var_df <-  train_data %>%
  select(one_of("ptid", cat_vars$var)) %>%
  pivot_longer(cols = cat_vars$var, names_to = "var", values_to = "value") %>%
  left_join(cat_vars, by = "var") %>%
  filter(!is.na(value)) %>%
  group_by(var, varlab, value) %>%
  summarise(n = n()) %>%
  group_by(varlab) %>%
  mutate(freq = n / sum(n)) %>%
  ungroup() %>%
  mutate(
    nudge_x = case_when(
      freq < 0.5 ~ 0.15,
      TRUE ~ -0.15
    )
  )

## `summarise()` regrouping output by 'var', 'varlab' (override with `.groups` argument)

ggplot(cat_var_df,
       aes(x = freq, y = value)) +
  geom_point() + 
  geom_text(aes(label = formatC(freq, format = "f", digits = 2)),
            nudge_x = cat_var_df$nudge_x, size = 3.5) +
  facet_wrap(~varlab, scales = "free_y", ncol = 4) +
  xlim(0, 1) +
  xlab("Proportion") +
  ylab("") +
  theme(axis.text.x = element_text(size = 10),
        axis.text.y = element_text(size = 10),
        strip.text.x = element_text(size = 7))

3.3.1.2 Continuous

3.3.1.2.1 Box plot

pivot_continuous_longer <- function(data, vars){
  col_names <- vars$var
  train_data %>%
    select(one_of("ptid", col_names)) %>%
    pivot_longer(cols = col_names, 
                 names_to = "var", 
                 values_to = "value") %>%
    left_join(vars, by = "var") %>%
    filter(!is.na(value)) 
}
continuous_var_df <- pivot_continuous_longer(train_data, 
                                             vars = continuous_vars)

plot_box <- function(data){
  ggplot(data, 
       aes(x = varlab, y = value)) +
  geom_boxplot(outlier.size = 1) + 
  facet_wrap(~varlab, scales = "free") +
  xlab("") +
  ylab("Value") +
   theme(axis.title.x=element_blank(),
        axis.text.x=element_blank(),
        axis.ticks.x=element_blank(),
        strip.text = element_text(size = 7))
}
plot_box(continuous_var_df)

3.3.1.2.2 Histogram

plot_hist <- function(data){
  ggplot(data,
       aes(x = value)) +
  geom_histogram(bins = 40, color = "white") +
  facet_wrap(~varlab, scales = "free", ncol = 4) + 
  xlab("") + ylab("Frequency") +
  theme(strip.text = element_text(size = 7))
}
plot_hist(continuous_var_df)

3.3.1.2.3 Outliers

Visual inspection of the box plots and histograms suggests that there are significant outliers in the labs. Let’s look at how many observations lie above the 99th percentile of the data and the “outer fence” (defined as the 3rd quartile plus 3 time the interquartile range). We will then create new lab variables truncated from above at the outer fence and replot the histograms.

outer_fence <- function(v){
  q1 <- quantile(v, .25, na.rm = TRUE)
  q3 <- quantile(v, .75, na.rm = TRUE)
  iq <- (q3 - q1)
  return(as.numeric(q3 + 3 * iq))
}

format_percent <- function(x){
  paste0(formatC(100 * x, format = "f", digits = 1), "%")
}

train_data %>%
  select(one_of(lab_vars$var)) %>%
  pivot_longer(cols = lab_vars$var, names_to = "Lab") %>%
  filter(!is.na(value)) %>%
  group_by(Lab) %>%
  summarise(Maximum = max(value),
            `99%` = quantile(value, .99), 
            `Outer fence` = outer_fence(value),
            `% above outer fence` = format_percent(mean(value > outer_fence(value)))) %>%
  mutate(Lab = get_var_labs(Lab)) %>%
  kable() %>%
  kable_styling()

## `summarise()` ungrouping output (override with `.groups` argument)

Lab	Maximum	99%	Outer fence	% above outer fence
Alanine aminotransferase (ALT)	3017.00	224.700000	130.00	3.4%
Aspartate aminotransferase (AST)	7000.01	336.000000	157.00	3.6%
Creatinine	27.35	10.880500	3.24	6.4%
C-reactive protein (CRP)	638.00	359.030000	458.00	0.2%
Fibrin D-Dimer	114475.00	19499.300000	4993.00	7.0%
Ferritin	100000.01	7746.620000	3648.00	3.4%
Lactate dehydrogenase (LDH)	14007.00	1262.440000	1050.00	1.9%
Lymphocyte count	120.35	3.740000	3.50	1.2%
Neutrophil count	82.40	18.900000	18.29	1.2%
Procalcitonin	753.66	30.122600	1.27	10.2%
Platelet count (PLT)	1213.00	527.860000	569.00	0.7%
Oxygen saturation	100.00	100.000000	110.00	0.0%
Troponin I	95.40	2.190225	0.17	8.1%
White blood cell count (WBC)	127.50	22.600000	21.70	1.2%

# Truncate labs using outer fence
truncate_max <- function(v) outer_fence(v)

for (i in 1:length(lab_vars$var)){
  original_var <- lab_vars$var[i]
  truncated_var <- paste0(original_var, "_t")
  truncated_max_i <- truncate_max(train_data[[original_var]])
  train_data <- train_data %>% mutate(
    !!truncated_var := ifelse(get(original_var) > truncated_max_i, 
                              truncated_max_i, 
                              get(original_var))
    )
}

After truncating the labs, the probability distributions appear more reasonable.

lab_vars_t <- lab_vars %>%
  mutate(var = paste0(var, "_t"))
vars <- bind_rows(vars, lab_vars_t) 

continuous_vars_t <- bind_rows(
  demographic_continuous_vars,
  vital_continuous_vars,
  lab_vars_t
)

continuous_var_t_df <- pivot_continuous_longer(train_data, 
                                               vars = continuous_vars_t)
plot_hist(continuous_var_t_df)

3.4 Bivariate relationships

3.4.1 Age and comorbidities

We start with the Charlson Comorbidity Index (CCI).

ggplot(train_data, aes_string(x = "age", y = "cci")) + 
  geom_point() +
  geom_smooth() +
  xlab(get_var_labs("age")) +
  ylab(get_var_labs("cci"))

Next, we will examine each comorbidity separately by plotting the probability that a patient has each comorbidity as a function of their age.

train_data[, c("age", comorbidity_cat_vars$var)] %>%
  pivot_longer(cols = comorbidity_cat_vars$var,
               names_to = "comorbidity") %>%
  mutate(value = ifelse(value == "No", 0L, 1L),
         comorbidity = get_var_labs(comorbidity)) %>%
  
  # Make plot
  ggplot(aes(x = age, y = value)) + 
    geom_smooth() +
    facet_wrap(~comorbidity, ncol = 4) +
   xlab(get_var_labs("age")) +
   ylab("Probability")

3.4.2 Age and vitals

plot_age_scatter_continuous <- function(vars){
  train_data[, c("age", vars)] %>%
  pivot_longer(cols = all_of(vars)) %>%
  mutate(name = get_var_labs(name)) %>%
  
  # Make plot
  ggplot(aes(x = age, y = value)) + 
    geom_smooth() +
    facet_wrap(~name, ncol = 4, scales = "free_y") +
   xlab(get_var_labs("age")) +
   ylab("Value")
}
plot_age_scatter_continuous(vital_continuous_vars$var)

## Warning: Removed 3863 rows containing non-finite values (stat_smooth).

3.4.3 Age and labs

plot_age_scatter_continuous(lab_vars_t$var) +
   theme(strip.text.x = element_text(size = 7))

## Warning: Removed 55407 rows containing non-finite values (stat_smooth).

3.4.4 Outcomes

train_data %>%
  count(died) %>%
  mutate(died = ifelse(died == 1, "Yes", "No"),
         prop = n/sum(n)) %>%
  ggplot(aes(x = died, y = prop)) +
  geom_bar(stat = "identity") +
  geom_text(aes(label = formatC(prop, format = "f", digits = 2)),
            nudge_y = .03, size = 3) +
  xlab("Died") +
  ylab("Proportion") 

3.4.4.1 Time to death

train_data %>%
  mutate(months_to_death = death_month - index_month) %>%
  filter(!is.na(months_to_death)) %>%
  group_by(months_to_death) %>%
  tally() %>%
  
  # Plot
  ggplot(aes(x = factor(months_to_death), y = n)) +
  geom_bar(stat = "identity", position = "dodge") +
  xlab("Months from index date to death") +
  ylab("Count")

3.5 Table 1

# Select variables for table 1
varnames_to_remove <- c("race", "ethnicity", "bmi_cat", "hypunc", "hypc", 
                        "diabunc", "diabwc")

all_varnames <- bind_rows(
  demographic_cat_vars,
  demographic_continuous_vars,
  comorbidity_cat_vars,
  comorbidity_continuous_vars,
  vital_cat_vars,
  vital_continuous_vars,
  lab_vars
) %>%
  filter(!var %in% varnames_to_remove) %>%
  pull(var)

categorical_varnames <- bind_rows(
  demographic_cat_vars,
  comorbidity_cat_vars,
  vital_cat_vars
) %>%
  filter(!var %in% varnames_to_remove) %>%
  pull(var)

non_normal_varnames <- c(
  demographic_continuous_vars$var,
  vital_continuous_vars$var,
  comorbidity_continuous_vars$var,
  lab_vars$var
)
  
## Create a TableOne object
table1 <- train_data %>%
  select(one_of(all_varnames, "died")) %>%
  rename_with(get_var_labs, .cols = all_of(all_varnames)) %>%
  rename(Survivor = died) %>%
  CreateTableOne(vars = get_var_labs(all_varnames), 
                 factorVars = get_var_labs(categorical_varnames), 
                 strata = "Survivor",  addOverall = TRUE, 
                 data = .)

## Print table 1
print(table1, nonnormal = get_var_labs(non_normal_varnames),  
      cramVars = c("Sex"),
      contDigits = 1, missing = TRUE, printToggle = FALSE) %>%
  set_colnames(c("Overall", "Survivor", "Non-survivor",
                "p", "Test", "Missing")) %>%
  kable() %>%
  kable_styling()

	Overall	Survivor	Non-survivor	p	Test	Missing
n	13658	11495	2163
Sex = Female/Male (%)	6563/7091 (48.1/51.9)	5635/5856 (49.0/51.0)	928/1235 (42.9/57.1)	<0.001		0.0
Geographic division (%)				<0.001		2.6
East North Central	4627 (34.8)	3954 (35.3)	673 (31.9)
East South Central	44 ( 0.3)	42 ( 0.4)	2 ( 0.1)
Middle Atlantic	4636 (34.9)	3844 (34.3)	792 (37.6)
Mountain	147 ( 1.1)	133 ( 1.2)	14 ( 0.7)
New England	1583 (11.9)	1272 (11.4)	311 (14.8)
Other	2 ( 0.0)	2 ( 0.0)	0 ( 0.0)
Pacific	511 ( 3.8)	438 ( 3.9)	73 ( 3.5)
South Atl/West South Crl	364 ( 2.7)	317 ( 2.8)	47 ( 2.2)
West North Central	1384 (10.4)	1189 (10.6)	195 ( 9.3)
Smoking (%)				<0.001		25.6
Current	866 ( 8.5)	785 ( 9.0)	81 ( 5.5)
Never	6207 (61.1)	5450 (62.8)	757 (51.1)
Previous	3092 (30.4)	2450 (28.2)	642 (43.4)
Race/Ethnicity (%)				<0.001		26.1
Non-Hispanic white	5647 (56.0)	4455 (53.1)	1192 (70.3)
Asian	362 ( 3.6)	307 ( 3.7)	55 ( 3.2)
Hispanic	533 ( 5.3)	478 ( 5.7)	55 ( 3.2)
Non-Hispanic black	3547 (35.2)	3153 (37.6)	394 (23.2)
Age (median [IQR])	62.0 [49.0, 75.0]	59.0 [46.0, 71.0]	77.0 [67.0, 85.0]	<0.001	nonnorm	0.0
Calendar time (median [IQR])	47.0 [38.0, 64.0]	47.0 [38.0, 64.0]	46.0 [37.0, 60.0]	<0.001	nonnorm	0.0
Acute myocardial infarction = Yes (%)	1535 (11.2)	1028 ( 8.9)	507 (23.4)	<0.001		0.0
Congestive heart failure = Yes (%)	2325 (17.0)	1604 (14.0)	721 (33.3)	<0.001		0.0
Peripheral vascular disease = Yes (%)	1671 (12.2)	1176 (10.2)	495 (22.9)	<0.001		0.0
Cerebrovascular disease = Yes (%)	1439 (10.5)	1023 ( 8.9)	416 (19.2)	<0.001		0.0
Dementia = Yes (%)	1394 (10.2)	854 ( 7.4)	540 (25.0)	<0.001		0.0
Chronic pulmonary disease = Yes (%)	3627 (26.6)	2933 (25.5)	694 (32.1)	<0.001		0.0
Rheumatoid disease = Yes (%)	398 ( 2.9)	315 ( 2.7)	83 ( 3.8)	0.007		0.0
Peptic ulcer disease = Yes (%)	206 ( 1.5)	160 ( 1.4)	46 ( 2.1)	0.013		0.0
Mild liver disease = Yes (%)	879 ( 6.4)	711 ( 6.2)	168 ( 7.8)	0.007		0.0
Hemiplegia or paraplegia = Yes (%)	330 ( 2.4)	228 ( 2.0)	102 ( 4.7)	<0.001		0.0
Renal disease = Yes (%)	2833 (20.7)	1984 (17.3)	849 (39.3)	<0.001		0.0
Cancer = Yes (%)	1678 (12.3)	1282 (11.2)	396 (18.3)	<0.001		0.0
Moderate/severe liver disease = Yes (%)	128 ( 0.9)	88 ( 0.8)	40 ( 1.8)	<0.001		0.0
Metastatic cancer = Yes (%)	277 ( 2.0)	188 ( 1.6)	89 ( 4.1)	<0.001		0.0
AIDS/HIV = Yes (%)	101 ( 0.7)	89 ( 0.8)	12 ( 0.6)	0.339		0.0
Diabetes = Yes (%)	4612 (33.8)	3669 (31.9)	943 (43.6)	<0.001		0.0
Hypertension = Yes (%)	8003 (58.6)	6333 (55.1)	1670 (77.2)	<0.001		0.0
CCI (median [IQR])	1.0 [0.0, 3.0]	1.0 [0.0, 2.0]	3.0 [1.0, 5.0]	<0.001	nonnorm	0.0
Body Mass Index (BMI) (median [IQR])	29.7 [25.5, 35.1]	30.0 [25.8, 35.4]	28.1 [24.0, 33.5]	<0.001	nonnorm	11.9
Diastolic blood pressure (median [IQR])	73.0 [65.5, 80.5]	74.0 [66.5, 81.0]	68.0 [60.0, 75.5]	<0.001	nonnorm	3.1
Systolic blood pressure (median [IQR])	126.0 [115.0, 139.0]	127.0 [116.0, 139.0]	122.0 [109.0, 136.5]	<0.001	nonnorm	3.2
Heart rate (median [IQR])	87.5 [77.5, 98.0]	87.0 [77.5, 98.0]	89.0 [77.5, 102.0]	<0.001	nonnorm	3.1
Respiration rate (median [IQR])	20.0 [18.0, 22.0]	19.5 [18.0, 21.0]	22.0 [19.0, 26.0]	<0.001	nonnorm	3.9
Temperature (median [IQR])	37.0 [36.7, 37.4]	37.0 [36.7, 37.4]	37.1 [36.7, 37.6]	<0.001	nonnorm	3.1
Alanine aminotransferase (ALT) (median [IQR])	28.0 [18.0, 46.0]	28.0 [18.0, 46.0]	27.0 [18.0, 44.0]	0.112	nonnorm	20.1
Aspartate aminotransferase (AST) (median [IQR])	37.0 [25.0, 58.0]	35.0 [25.0, 54.0]	46.0 [30.0, 73.0]	<0.001	nonnorm	21.0
C-reactive protein (CRP) (median [IQR])	79.1 [34.0, 140.0]	72.2 [30.0, 130.0]	116.0 [63.0, 184.0]	<0.001	nonnorm	38.7
Creatinine (median [IQR])	1.0 [0.8, 1.4]	1.0 [0.8, 1.3]	1.3 [1.0, 2.1]	<0.001	nonnorm	10.4
Ferritin (median [IQR])	510.0 [224.0, 1080.0]	470.0 [207.0, 992.0]	747.5 [320.8, 1501.5]	<0.001	nonnorm	43.6
Fibrin D-Dimer (median [IQR])	750.0 [390.0, 1540.8]	692.5 [370.0, 1346.5]	1345.0 [668.2, 3315.0]	<0.001	nonnorm	90.4
Lactate dehydrogenase (LDH) (median [IQR])	321.0 [238.0, 441.0]	308.0 [232.0, 415.0]	404.0 [284.0, 556.5]	<0.001	nonnorm	45.2
Lymphocyte count (median [IQR])	1.0 [0.7, 1.4]	1.0 [0.7, 1.4]	0.8 [0.5, 1.1]	<0.001	nonnorm	11.2
Neutrophil count (median [IQR])	4.9 [3.4, 7.1]	4.7 [3.2, 6.7]	6.1 [4.1, 9.2]	<0.001	nonnorm	11.2
Oxygen saturation (median [IQR])	96.0 [94.0, 98.0]	96.0 [94.5, 98.0]	95.0 [93.0, 97.0]	<0.001	nonnorm	3.9
Procalcitonin (median [IQR])	0.1 [0.1, 0.4]	0.1 [0.1, 0.3]	0.3 [0.1, 1.0]	<0.001	nonnorm	49.3
Troponin I (median [IQR])	0.0 [0.0, 0.0]	0.0 [0.0, 0.0]	0.0 [0.0, 0.1]	<0.001	nonnorm	41.2
Platelet count (PLT) (median [IQR])	202.0 [157.0, 260.0]	205.0 [160.0, 262.0]	187.5 [143.0, 245.0]	<0.001	nonnorm	9.8
White blood cell count (WBC) (median [IQR])	6.7 [4.9, 9.1]	6.5 [4.8, 8.7]	7.7 [5.6, 11.1]	<0.001	nonnorm	9.7

#summary(table1)

3.6 Transformations of continuous variables

The functional form of the relationship between mortality and the continuous variables is assessed using a series of univariate fits. We mainly rely on visual inspection of the graphs but also report the Bayesian information criterion (BIC).

fit_univariate_logit <- function(var, data){
  make_f <- function(rhs){
    as.formula(paste("died", rhs, sep =" ~ "))
  }
  fit_logit <- function(f, data){
    glm(f, family = "binomial", data = data)
  }
  
  list(
    `Linear` = fit_logit(make_f(var), data),
    `Spline 3 knots` = fit_logit(make_f(sprintf("ns(%s, 5)", var)), data),
    `Spline 4 knots` = fit_logit(make_f(sprintf("ns(%s, 6)", var)), data),
    `Spline 5 knots` = fit_logit(make_f(sprintf("ns(%s, 7)", var)), data)
  )
}

predict_univariate_logit <- function(models, var, var_values, type = "response"){
  newdata <- data.frame(var = var_values)
  colnames(newdata) <- var
  map_dfc(models,
         function(x) predict(x, newdata = newdata, type = type)) %>%
  mutate(var = var_values) %>%
  pivot_longer(cols = names(models),
               names_to = "Model",
               values_to = "y")
}

midpoint <- function(x, digits = 2){
  lower <- as.numeric(gsub(",.*", "", gsub("\\(|\\[|\\)|\\]", "", x)))
  upper <- as.numeric(gsub(".*,", "", gsub("\\(|\\[|\\)|\\]", "", x)))
  return(round(lower+(upper-lower)/2, digits))
}

bin_y <- function(var, var_values){
  data <- train_data[, c(var, "died")] %>%
    filter(!is.na(get(var)))
  data <- data %>%
    mutate(x_cat = cut(get(var), breaks = 20),
           x_midpoint = midpoint(x_cat)) %>%
    group_by(x_midpoint) %>%
    summarise(y = mean(died),
              n = n())
  colnames(data)[1] <- var
  return(data)
}

plot_univariate_logit <- function(models, var, var_values, var_lab = "Variable",
                                  type = "response", ylab = "Probability of death"){
  # Plotting data
  predicted_probs <- predict_univariate_logit(models, var, var_values, type = type)
  ylab <- switch(type,
                 "link" = "Log odds",
                 "response" = "Probability of death",
                 stop("Type must be 'link' or 'response'")
  )
  binned_y <- bin_y(var, var_values)
  if (type == "link"){
    binned_y$y <- ifelse(binned_y$y == 0, .001, binned_y$y)
    binned_y$y <- ifelse(binned_y$y == 1, .99, binned_y$y)
    binned_y$y <- qlogis(binned_y$y)
  }
  
  # Plotting scales
  y_min <- min(c(binned_y$y, predicted_probs$y))
  y_max <- max(c(binned_y$y, predicted_probs$y))
  size_breaks <- seq(min(binned_y$n), max(binned_y$n),
                     length.out = 6)
  
  # Plot
  ggplot(predicted_probs,
       aes(x = var, y = y)) +
    geom_line() +
    geom_point(data = binned_y, aes_string(x = var, y = "y", size = "n")) +
    facet_wrap(~Model, ncol = 2) +
    xlab(var_lab) +
    ylab(ylab) +
    ylim(floor(y_min), ceiling(y_max)) +
    scale_size(name = "Sample size", range = c(0.3, 3), 
               breaks = round(size_breaks, 0)) 
}

make_seq <- function(var){
  var_min <- min(train_data[[var]], na.rm = TRUE)
  var_max <- max(train_data[[var]], na.rm = TRUE)
  seq(var_min, var_max, length.out = 100)
}

evaluate_univariate_logit <- function(var, print = TRUE){
  var_values = make_seq(var)
  var_lab = get_var_labs(var)
  
  # Do evaluations
  fits <- fit_univariate_logit(var, data = train_data)
  p_link <- plot_univariate_logit(fits, var, var_values, var_lab, type = "link")
  p_probs <- plot_univariate_logit(fits, var, var_values, var_lab, type = "response")
  bic <- sapply(fits, BIC)
  
  # Print and return
  if (print){
    print(p_link)
    print(p_probs)
    print(bic)
  }
  return(list(fits = fits, p_link = p_link, p_probs = p_probs,
              bic = bic))
}

3.6.1 Demographics

3.6.1.1 Age

uv_age <- evaluate_univariate_logit("age")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10153.39       10179.96       10189.56       10198.91

3.6.1.2 Calendar time

uv_calendar_time <- evaluate_univariate_logit("calendar_time")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11927.48       11947.86       11956.50       11965.07

3.6.2 Vitals

3.6.2.1 Body Mass Index

uv_bmi <- evaluate_univariate_logit("bmi")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10634.18       10628.07       10636.99       10642.34

3.6.2.2 Diastolic blood pressure

uv_dbp <- evaluate_univariate_logit("dbp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11072.71       11064.10       11073.56       11082.72

3.6.2.3 Systolic blood pressure

uv_sbp <- evaluate_univariate_logit("sbp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11463.15       11244.37       11253.16       11262.85

3.6.2.4 Heart rate

uv_hr <- evaluate_univariate_logit("hr")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11501.00       11406.44       11414.25       11422.72

3.6.2.5 Respiration rate

uv_resp <- evaluate_univariate_logit("resp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10747.57       10511.21       10520.03       10528.45

3.6.2.6 Temperature

uv_temp <- evaluate_univariate_logit("temp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11594.91       11352.92       11360.11       11367.23

3.6.3 Labs

3.6.3.1 Alanine aminotransferase (U/L)

uv_alt <- evaluate_univariate_logit("alt")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10129.18       10159.31       10166.55       10172.47

uv_alt_t <- evaluate_univariate_logit("alt_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10127.15       10161.10       10170.62       10178.92

3.6.3.2 Alanine aminotransferase (U/L)

uv_ast <- evaluate_univariate_logit("ast")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       9998.672       9837.378       9844.663       9852.407

uv_ast_t <- evaluate_univariate_logit("ast_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       9850.176       9840.492       9849.651       9858.934

3.6.3.3 C-reactive protein (mg/L)

uv_crp <- evaluate_univariate_logit("crp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7611.060       7598.858       7607.186       7613.877

uv_crp_t <- evaluate_univariate_logit("crp_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7609.387       7598.807       7607.153       7613.778

3.6.3.4 Creatinine (mg/dL)

uv_creatinine <- evaluate_univariate_logit("creatinine")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10993.75       10514.41       10503.31       10504.14

uv_creatinine_t <- evaluate_univariate_logit("creatinine_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10601.63       10480.42       10488.51       10495.10

3.6.3.5 Ferritin (ng/mL)

uv_ferritin <- evaluate_univariate_logit("ferritin")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7309.122       7198.348       7208.925       7215.448

uv_ferritin_t <- evaluate_univariate_logit("ferritin_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7214.138       7200.565       7211.500       7216.768

3.6.3.6 Fibrin D-Dimer (ng/mL)

uv_d_dimer <- evaluate_univariate_logit("d_dimer")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       1201.612       1189.221       1193.070       1197.217

uv_d_dimer_t <- evaluate_univariate_logit("d_dimer_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       1154.281       1177.857       1184.975       1192.032

3.6.3.7 Lactate dehydrogenase (U/L)

uv_ldh <- evaluate_univariate_logit("ldh")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       6916.380       6727.567       6730.615       6736.431

uv_ldh_t <- evaluate_univariate_logit("ldh_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       6722.740       6711.902       6720.931       6729.386

3.6.3.8 Lymphocyte count (10^3/uL)

uv_lymphocyte <- evaluate_univariate_logit("lymphocyte")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11021.34       10676.61       10684.69       10693.36

uv_lymphocyte_t <- evaluate_univariate_logit("lymphocyte_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10839.18       10669.88       10677.76       10686.35

3.6.3.9 Neutrophil count (10^3/uL)

uv_neutrophil <- evaluate_univariate_logit("neutrophil")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10741.03       10722.67       10723.14       10730.54

uv_neutrophil_t <- evaluate_univariate_logit("neutrophil_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10709.62       10710.56       10716.35       10724.36

3.6.3.10 Oxygen saturation (%)

uv_spo2 <- evaluate_univariate_logit("spo2")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11141.19       11109.15       11118.68       11128.31

3.6.3.11 Procalcitonin (ng/mL)

uv_pct <- evaluate_univariate_logit("pct")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       6956.604       6495.873       6495.437       6501.271

uv_pct_t <- evaluate_univariate_logit("pct_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       6565.428       6491.765       6493.818       6502.947

3.6.3.12 Troponin I (ng/mL)

uv_tni <- evaluate_univariate_logit("tni")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7750.579       7043.096       7048.626       7057.567

uv_tni_t <- evaluate_univariate_logit("tni_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7074.305       6951.225       6959.737       6968.103

3.6.3.13 Platelet count (10^3/uL)

uv_plt <- evaluate_univariate_logit("plt")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11102.84       11079.15       11086.55       11095.10

uv_plt_t <- evaluate_univariate_logit("plt_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11099.10       11078.61       11086.35       11094.90

3.6.3.14 White blood cell count (10^3/uL)

uv_wbc <- evaluate_univariate_logit("wbc")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10987.01       10940.40       10944.79       10951.35

uv_wbc_t <- evaluate_univariate_logit("wbc_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10934.25       10925.33       10933.11       10940.05

4 Data preprocessing

4.1 Candidate variables for modeling

Let’s specify variables that will be candidates for our model and used during variable selection. We will combine race and ethnicity into a single variable, but will wait to do this until after multiple imputation (see next section).

demographics_to_include <- c("age", "sex", "race", "ethnicity", "division",
                             "smoke", "calendar_time")
comorbidities_to_include <- comorbidity_cat_vars %>%
  filter(!var %in% c("diabunc", "diabwc", "hypunc", "hypc")) %>%
  pull(var) %>%
  c("diab", "hyp")
vitals_to_include <- c("bmi", "temp", "hr", "resp", "sbp")
labs_to_include <- c("spo2", "crp_t", "tni_t", "ast_t", "ferritin_t",
                     "creatinine_t", "ldh_t", "lymphocyte_t", "neutrophil_t",
                     "plt_t", "wbc_t")
vars <- vars %>%
  mutate(include = ifelse(var %in% c(demographics_to_include, 
                                     comorbidities_to_include,
                                     vitals_to_include, 
                                     labs_to_include),
                          1, 0))
get_included_vars <- function(){
  vars[vars$include == 1, ]$var
}

make_rhs <- function(vars){
  as.formula(paste0("~", paste(vars, collapse = " + ")))
}
candidate_model_rhs <- make_rhs(get_included_vars()) 

4.2 Missing data imputation

Note that there is no “Other” geographic region as all 9 of the US geographic divisions are included in the data. We will consequently recode this category to missing.

train_data <- train_data %>%
  mutate(division = ifelse(division == "Other", NA, division),
         division = ifelse(division %in% c("East South Central", "Mountain"),
                           "Other", division))

Let’s re-level them so that we have preferred reference categories.

train_data <- train_data %>% mutate(
  sex = relevel(factor(sex), ref = "Male"),
  division = relevel(factor(division), ref = "Pacific"),
  smoke = relevel(factor(smoke), ref = "Never"),
  bmi_cat = relevel(factor(bmi_cat), ref = "Normal")
)

We can now multiply impute data use MICE.

# Run MICE algorithm
mice_out <- train_data %>%
  select(c(one_of(get_included_vars()), "died")) %>%
  mutate_if(is.character, as.factor) %>%
  mice(m = n_imputations, maxit = 5) 

## 
##  iter imp variable
##   1   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   1   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   2   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   2   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   3   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   3   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   4   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   4   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   5   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   5   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t

# Append datasets and add outcome
mi_df <- complete(mice_out, action = "long", include = TRUE) %>%
  as_tibble() %>%
  mutate(died = rep(train_data$died, mice_out$m + 1))

# To compare MICE to aregImpute
# areg_out <-  aregImpute(update.formula(candidate_model_rhs, ~.), 
#                                        n.impute = 2, data = train_data)

4.3 Imputation diagnostics

The distributions of the imputed and observed data are compared as a diagnostic for the imputation. They look pretty similar suggesting that there is nothing terribly wrong with the imputation.

make_imp_df <- function(object){
  # Get imputations
  if (inherits(object, "mids")){
    imp <- object$imp
  } else{ # aregImpute
   imp <- object$imputed 
   for (i in 1:length(imp)){
     cat_levels_i <- object$cat.levels[[i]]
     if (!is.null(cat_levels_i) && !is.null(imp[[i]])){
       levels <- sort(unique(c(imp[[i]])))
       imp[[i]] <- apply(imp[[i]], 
                         2, 
                         function(x) factor(x, levels = levels,
                                            labels = cat_levels_i))
     }
   }
  }
  
  # Create list of data frames
  is_numeric <- sapply(imp, function (x) is.numeric(x[, 1]))
  continuous_df <- vector(mode = "list", length = sum(is_numeric))
  cat_df <- vector(mode = "list", length = sum(!is_numeric))
  continuous_cntr <- 1
  cat_cntr <- 1
  for (i in 1:length(imp)){
    if(!is.null(nrow(imp[[i]])) && nrow(imp[[i]]) > 0 ){
        imp_i_df <- data.frame(var = names(imp)[i],
                               imp = rep(1:ncol(imp[[i]]), each = nrow(imp[[i]])),
                               value = c(as.matrix(imp[[i]]))) %>% 
          as_tibble()
        
    } else{
      imp_i_df <- NULL
    }
    if (is_numeric[i]){
      continuous_df[[continuous_cntr]] <- imp_i_df
      continuous_cntr <- continuous_cntr + 1
    } else{
      cat_df[[cat_cntr]] <- imp_i_df
      cat_cntr <- cat_cntr + 1
    }
  } 
  
  # Row bind data frames
  continuous_df = bind_rows(continuous_df) %>%
    mutate(obs = "Imputed",
           varlab = get_var_labs(var)) 
  
  cat_df = bind_rows(cat_df) %>%
    mutate(obs = "Imputed",
           varlab = get_var_labs(var)) 
              
  # Return
  return(list(continuous = continuous_df,
              cat = cat_df))
}
imp_df <- make_imp_df(mice_out)
#imp_df <- make_imp_df(areg_out)

# Plot continuous variables
## Data for plotting
obsimp_df_continuous <- bind_rows(
  imp_df$continuous,
  continuous_var_df %>%
    select(var, value, varlab) %>%
    mutate(imp = 0, obs = "Observed")
) %>%
  mutate(imp = ifelse(imp == 0, "Observed", paste0("Imputation ", imp))) %>%
  filter(var %in% unique(imp_df$continuous$var))
  
## Plot
ggplot(obsimp_df_continuous,
       aes(x = value, col = imp)) +
  geom_density(position = "jitter") +
  facet_wrap(~varlab, scales = "free", ncol = 3) +
  xlab("") + ylab("Density") +
  scale_color_discrete(name = "") +
  theme(legend.position = "bottom")

# Plot categorical variables
## Data for plotting
obsimp_df_cat <- 
  bind_rows(
    imp_df$cat %>%
      group_by(var, varlab, value, imp) %>%
      summarise(n = n()) %>%
      group_by(var, varlab, imp) %>%
      mutate(freq = n / sum(n)),
    cat_var_df %>%
      select(var, value, varlab, n, freq) %>%
      mutate(imp = 0, obs = "Observed")
  ) %>%
    mutate(imp = ifelse(imp == 0, "Observed", paste0("Imputation ", imp))) %>%
    filter(var %in% unique(imp_df$cat$var))

## `summarise()` regrouping output by 'var', 'varlab', 'value' (override with `.groups` argument)

# Plot
ggplot(obsimp_df_cat, 
       aes(x = value, y = freq, fill = imp)) +
  geom_bar(position = "dodge", stat = "identity") +
  facet_wrap(~varlab, scales = "free_x") +
  scale_fill_discrete(name = "") +
  xlab("") +
  ylab("Proportion") +
  theme(legend.position = "bottom",
        axis.text.x = element_text(angle = 45, vjust = 1, hjust = 1))

4.4 Recoding categorical variables

We create the race/ethnicity variable after imputation of missing values of race and ethnicity. We will also combine the diabetes and hypertension variables.

mi_df <- mi_df %>% 
  add_race_ethnicity() 

prop.table(table(mi_df %>% filter(.imp == 0) %>% pull(diab)))

## 
##        No       Yes 
## 0.6623224 0.3376776

prop.table(table(mi_df %>% filter(.imp == 0) %>% pull(hyp)))

## 
##        No       Yes 
## 0.4140431 0.5859569

Now lets use the combined race and ethnicity category in our model.

vars <- vars %>%
  mutate(include = case_when(
      var == "race_ethnicity" ~ 1,
      var == "race" ~ 0,
      var == "ethnicity" ~ 0,
      TRUE ~ include
    )
  )
  
candidate_model_rhs <- make_rhs(get_included_vars())

Finally, we will (i) add the new race/ethnicity variable to the “MICE” object and (ii) create a list of imputed datasets for analysis.

mice_out <- as.mids(mi_df)
mi_list <- mi_df %>% 
  filter(.imp > 0) %>% 
  split(list(.$.imp))

5 Variable selection

The continuous variables in our model are transformed using restricted cubic splines with 3 knots.

candidate_model_rhs <- candidate_model_rhs %>%
  update.formula( ~. + rcs(age, 3) - age +
                       rcs(calendar_time, 3) - calendar_time +
                       rcs(bmi, 3) - bmi +
                       rcs(temp, 3) - temp +
                       rcs(hr, 3) - hr +
                       rcs(resp, 3) - resp +
                       rcs(sbp, 3) - sbp +
                       rcs(spo2, 3) - spo2 +
                       rcs(crp_t, 3) - crp_t +
                       rcs(tni_t, 3) - tni_t +
                       rcs(ast_t, 3) - ast_t +
                       rcs(creatinine_t, 3) - creatinine_t +
                       rcs(ferritin_t, 3) - ferritin_t +
                       rcs(ldh_t, 3) - ldh_t,
                       rcs(lymphocyte_t, 3) - lymphocyte_t +
                       rcs(neutrophil_t, 3) - neutrophil_t +
                       rcs(plt_t, 3) - plt_t +
                       rcs(wbc_t, 3) - wbc_t)

With oem we need to create an x matrix since there is no formula interface.

rename_rcs <- function(v){
  rcs_ind <- grep("rcs", v)
  v[rcs_ind] <- sub("rcs.*)", "", v[rcs_ind])
  return(v)
}

rename_terms <- function(v){
  v <- gsub(" ", "_", v)
  v <- gsub("-", "", v)
  v <- gsub("/", "", v)
  v <- rename_rcs(v)
  return(v)
}

make_x <- function(data){
  x <- model.matrix(candidate_model_rhs, data)
  assign <- attr(x, "assign")
  colnames(x) <- rename_terms(colnames(x))
  x <- x[, -1]
  attr(x, "assign") <- assign[-1]
  return(x)
}

# List of x and y for each imputed dataset
x <- mi_list %>% map(make_x)
y <- mi_list %>% map(function(x) x[["died"]])

To make the graphs look nice, we will import labels for the model terms.

terms <- read.csv("risk-factors-terms.csv") %>%
  left_join(vars[, c("var", "group")], by = "var")

get_term_labs <- function(v, term_name = "term"){
  terms$termlab[match(v, terms[[term_name]])]
}

match_terms_to_vars <- function(t){
  terms$var[match(t, terms$term)]
}

Finally, we run the grouped lasso.

# Number of folds for cross-validation
n_folds <- 10

# Threshold for variable inclusion
inclusion_threshold <- 0.9

# Matrix to store inclusion results
inclusion_sim <- matrix(0,  ncol = ncol(x[[1]]) + 1, 
                        nrow = n_rep * n_imputations)

# Groups
groups <- attr(x[[1]], "assign") 

# Convenience function to extract coefficients from Group-Lasso
coef_cv_oem <- function(object){
  coef <- object$oem.fit$beta$grp.lasso
  lse_ind <- which(object$lambda[[1]] == object$lambda.1se.models)
  return(coef[, lse_ind])
}

# Variable selection via Group-Lasso
cntr <- 1
for (i in 1:n_imputations){
  for (j in 1:n_rep){
    
    # Cross-validation
    cv_fit <-  cv.oem(x = x[[i]], y = y[[i]], 
                      penalty = "grp.lasso", 
                      groups = groups, 
                      family = "binomial",
                      type.measure = "deviance",
                      nfolds = n_folds
                      )
  
    # Count nonzero coefficients 
    inclusion_sim[cntr, which(coef_cv_oem(cv_fit) != 0)] <- 1
    
    # Iterate
    cntr <- cntr + 1
  } # End repeated CV loop
} # End imputation loop 

Let’s check the inclusion probabilities from the above repeated cross-validation steps.

# Percentage of simulations each term is included
inclusion_sim <- inclusion_sim[, -1] # Remove intercept
colnames(inclusion_sim) = colnames(x[[1]])

inclusion_summary <- tibble(term = colnames(inclusion_sim), 
                            prob = apply(inclusion_sim, 2, mean)) 
model_terms <- inclusion_summary %>%
  filter(prob >= inclusion_threshold) %>%
  pull(term)

# Percentage of simulations each variable is included
inclusion_summary <- inclusion_summary %>%
  mutate(var = match_terms_to_vars(term)) %>%
  mutate(varlab = get_var_labs(var)) %>% 
  distinct(prob, var, varlab)

# Plot
ggplot(inclusion_summary,
       aes(x = reorder(varlab, prob), y = prob)) +
  geom_bar(stat = "identity") + 
  geom_hline(yintercept = inclusion_threshold, linetype = "dotted", 
                color = "red", size = 1) +
  ylab("Probability of inclusion") +
  coord_flip() +
  theme(axis.title.y = element_blank())

6 Model interpretation

6.1 Model terms and data

Variables are included in the model based on the group Lasso simulation implemented above.

vars_to_exclude <- inclusion_summary %>%
  filter(prob < inclusion_threshold) %>%
  pull(var)

remove_terms_from_rhs <- function(f, vars_to_exclude){
  # First convert formula to string separated by +
  f_string <- Reduce(paste, deparse(f))
  f_string <- gsub("~", "", f_string)
  f_string <- gsub(" ", "", f_string)
  
  # Then convert string to vector
  f_vec <-   unlist(strsplit(f_string, "\\+"))
  pattern_to_exclude <- paste(vars_to_exclude, collapse = "|")
  f_vec <- f_vec[!grepl(pattern_to_exclude, f_vec)]
  
  # Convert string back to formula
  f_new <- paste0("~", paste(f_vec, collapse = " + "))
  return(as.formula(f_new))
}

model_rhs <- remove_terms_from_rhs(candidate_model_rhs, vars_to_exclude)

label(mi_df) <- map(colnames(mi_df), 
                    function(x) label(mi_df[, x]) <- get_var_labs(x))
dd <- datadist(mi_df, adjto.cat = "first")
options(datadist = "dd")

6.2 Fit

As described in the paper, five models are fit that include the following predictors: (i) age only, (ii) comorbidities only, (iii) all demographics (and calendar time), and (iv) demographics (and calendar time) and comorbidities, and (v) all variables selected by the grouped lasso.

# The four models
lrm_names <- c("Age only",
               "Comorbidities only",
               "All demographics",
               "Demographics and comorbidities",
               "All variables")

## (1): fit_age: Only includes age
f_lrm_age <- died ~ age 
lrm_fit_age <- fit.mult.impute(f_lrm_age, fitter = lrm, xtrans = mice_out, pr = FALSE,
                               x = TRUE, y = TRUE)

## (2): fit_c: Only includes comorbidities
c_vars <- vars %>%
  filter(group == "Comorbidities" & include == 1 & !var %in% vars_to_exclude) %>%
  pull(var)
f_lrm_c <- as.formula(paste0("died ~", paste(c_vars, collapse = "+")))
lrm_fit_c <- fit.mult.impute(f_lrm_c, fitter = lrm, xtrans = mice_out, 
                             pr = FALSE, x = TRUE, y = TRUE) 

## (3): fit_d: All demographics including age
f_lrm_d <- update.formula(f_lrm_age, ~. + sex + rcs(calendar_time, 3) + 
                          race_ethnicity + division)
lrm_fit_d <- fit.mult.impute(f_lrm_d, fitter = lrm, xtrans = mice_out, pr = FALSE,
                             x = TRUE, y = TRUE)

## (4): fit_dc: Demographics and comorbidities
f_lrm_dc <- update.formula(f_lrm_d, 
                           as.formula(paste0("~.+", paste(c_vars, collapse = "+"))))
lrm_fit_dc <- fit.mult.impute(f_lrm_dc, fitter = lrm, xtrans = mice_out, pr = FALSE,
                              x = TRUE, y = TRUE)

## (5): fit_all: The main model including demographics, comorbidities, vitals, and labs
f_lrm_all <- update.formula(model_rhs, died ~ .)
lrm_fit_all <- fit.mult.impute(f_lrm_all, fitter = lrm, xtrans = mice_out, 
                               pr = FALSE, x = TRUE, y = TRUE)

### Note that we can also estimate the models with stats::glm
glm_fits_all <- mi_list %>%
  map(function (x) glm(f_lrm_all, data = x, family = "binomial")) 
glm_fit_all <- glm_fits_all %>% pool()

We will print the results of the full model for the interested reader

lrm_fit_all

## Logistic Regression Model
##  
##  fit.mult.impute(formula = f_lrm_all, fitter = lrm, xtrans = mice_out, 
##      pr = FALSE, x = TRUE, y = TRUE)
##  
##                         Model Likelihood    Discrimination    Rank Discrim.    
##                               Ratio Test           Indexes          Indexes    
##  Obs         13658    LR chi2    4198.88    R2       0.454    C       0.889    
##   0          11495    d.f.            58    g        2.383    Dxy     0.778    
##   1           2163    Pr(> chi2) <0.0001    gr      10.837    gamma   0.778    
##  max |deriv| 1e-09                          gp       0.205    tau-a   0.207    
##                                             Brier    0.088                     
##  
##                                    Coef     S.E.    Wald Z Pr(>|Z|)
##  Intercept                          14.7596  2.9659  4.98  <0.0001 
##  division=East North Central        -0.4597  0.1649 -2.79  0.0053  
##  division=Middle Atlantic           -0.3568  0.1654 -2.16  0.0310  
##  division=New England               -0.2798  0.1964 -1.42  0.1544  
##  division=Other                     -0.7914  0.5439 -1.45  0.1457  
##  division=South Atl/West South Crl  -0.6196  0.2401 -2.58  0.0099  
##  division=West North Central        -0.4910  0.1854 -2.65  0.0081  
##  smoke=Current                       0.0395  0.1306  0.30  0.7625  
##  smoke=Previous                      0.2213  0.0720  3.08  0.0021  
##  race_ethnicity=Asian               -0.0860  0.1911 -0.45  0.6527  
##  race_ethnicity=Hispanic            -0.1738  0.1133 -1.53  0.1251  
##  race_ethnicity=Non-Hispanic black  -0.5324  0.1099 -4.84  <0.0001 
##  ami=Yes                             0.0366  0.0892  0.41  0.6819  
##  chf=Yes                             0.1892  0.0770  2.46  0.0140  
##  pvd=Yes                             0.0699  0.0864  0.81  0.4186  
##  cevd=Yes                            0.0871  0.0907  0.96  0.3372  
##  dementia=Yes                        0.4110  0.0868  4.74  <0.0001 
##  cpd=Yes                             0.0291  0.0708  0.41  0.6813  
##  rheumd=Yes                          0.0501  0.1558  0.32  0.7479  
##  mld=Yes                             0.0595  0.1207  0.49  0.6222  
##  hp=Yes                              0.7147  0.1612  4.43  <0.0001 
##  rend=Yes                            0.0921  0.0863  1.07  0.2856  
##  canc=Yes                            0.0324  0.0850  0.38  0.7027  
##  msld=Yes                            0.8135  0.2574  3.16  0.0016  
##  metacanc=Yes                        0.8577  0.1788  4.80  <0.0001 
##  aids=Yes                            0.6035  0.3588  1.68  0.0925  
##  diab=Yes                            0.1188  0.0658  1.81  0.0707  
##  hyp=Yes                             0.1187  0.0786  1.51  0.1309  
##  lymphocyte_t                       -0.2275  0.0567 -4.01  <0.0001 
##  plt_t                              -0.0025  0.0004 -6.55  <0.0001 
##  wbc_t                               0.0476  0.0092  5.20  <0.0001 
##  age                                 0.0670  0.0046 14.53  <0.0001 
##  age'                               -0.0051  0.0055 -0.92  0.3582  
##  calendar_time                      -0.0169  0.0046 -3.68  0.0002  
##  calendar_time'                      0.0073  0.0078  0.94  0.3476  
##  bmi                                -0.0278  0.0116 -2.40  0.0163  
##  bmi'                                0.0475  0.0136  3.49  0.0005  
##  temp                               -0.2057  0.0676 -3.04  0.0023  
##  temp'                               0.9482  0.1138  8.33  <0.0001 
##  hr                                  0.0029  0.0044  0.65  0.5131  
##  hr'                                 0.0099  0.0049  2.03  0.0423  
##  resp                                0.0341  0.0216  1.58  0.1145  
##  resp'                               0.0513  0.0245  2.10  0.0360  
##  sbp                                -0.0325  0.0038 -8.61  <0.0001 
##  sbp'                                0.0302  0.0043  7.02  <0.0001 
##  spo2                               -0.1268  0.0168 -7.55  <0.0001 
##  spo2'                               0.1271  0.0283  4.49  <0.0001 
##  crp_t                               0.0028  0.0014  2.01  0.0444  
##  crp_t'                             -0.0029  0.0018 -1.59  0.1127  
##  tni_t                              16.2209  4.3080  3.77  0.0002  
##  tni_t'                            -46.0455 18.2946 -2.52  0.0118  
##  ast_t                               0.0061  0.0040  1.50  0.1324  
##  ast_t'                             -0.0133  0.0068 -1.94  0.0525  
##  creatinine_t                        0.3504  0.1187  2.95  0.0032  
##  creatinine_t'                      -0.2678  0.1923 -1.39  0.1638  
##  ferritin_t                          0.0001  0.0002  0.58  0.5633  
##  ferritin_t'                        -0.0001  0.0004 -0.30  0.7662  
##  ldh_t                               0.0051  0.0010  4.97  <0.0001 
##  ldh_t'                             -0.0041  0.0013 -3.17  0.0015  
## 

6.3 Collinearity

Let’s examine the extent to which our predictors are collinear: for categorical and continous variables, we use an anova model; for categorical and categorical variables, we use Cramer’s V; and for continous and continous variables, we use spearman correlation. (Note that this takes a long time to run and could probablty be made more efficient.)

## from https://stackoverflow.com/questions/52554336/plot-the-equivalent-of-correlation-matrix-for-factors-categorical-data-and-mi
mixed_assoc <- function(df, cor_method = "spearman", adjust_cramersv_bias = TRUE){
  df_comb <- expand.grid(names(df), names(df),  stringsAsFactors = F) %>% 
    set_names("X1", "X2")
  is_nominal <- function(x) inherits(x, c("factor", "character"))
  # https://community.rstudio.com/t/why-is-purr-is-numeric-deprecated/3559
  # https://github.com/r-lib/rlang/issues/781
  is_numeric <- function(x) { is.integer(x) || is_double(x)}
  f <- function(x_name, y_name) {
    x <-  pull(df, x_name)
    y <-  pull(df, y_name)
    result <- if(is_nominal(x) && is_nominal(y)){
        # use bias corrected cramersV as described in https://rdrr.io/cran/rcompanion/man/cramerV.html
        cv <- cramerV(as.character(x), as.character(y), 
                      bias.correct = adjust_cramersv_bias)
        data.frame(x_name, y_name, assoc = cv, type = "cramersV")
    } else if(is_numeric(x) && is_numeric(y)){
        correlation <- cor(x, y, method = cor_method, use = "complete.obs")
        data.frame(x_name, y_name, assoc = correlation, type = "correlation")
    } else if(is_numeric(x) && is_nominal(y)){
        # from https://stats.stackexchange.com/questions/119835/correlation-between-a-nominal-iv-and-a-continuous-dv-variable/124618#124618
        r_squared <- summary(lm(x ~ y))$r.squared
        data.frame(x_name, y_name, assoc = sqrt(r_squared), type = "anova")
    } else if(is_nominal(x) && is_numeric(y)){
        r_squared <- summary(lm(y ~ x))$r.squared
        data.frame(x_name, y_name, assoc = sqrt(r_squared), type = "anova")
    } else {
        warning(paste("unmatched column type combination: ", class(x), class(y)))
    }
    # finally add complete obs number and ratio to table
    result %>% 
      mutate(
        complete_obs_pairs = sum(!is.na(x) & !is.na(y)),
        complete_obs_ratio = complete_obs_pairs/length(x)) %>%
      rename(x = x_name, y = y_name)
  }
  # apply function to each variable combination
  map2_df(df_comb$X1, df_comb$X2, f)
}

# Create correlation matrix of associations
corr_mat <- mi_df %>% 
  filter(.imp == 1) %>% 
  select(any_of(get_included_vars())) %>%
  mixed_assoc() %>%
  select(x, y, assoc) %>%
  pivot_wider(names_from = y, values_from = assoc) %>%
  column_to_rownames("x") %>%
  as.matrix

# Make tile plot
m <- abs(corr_mat)
heatmap_df <- tibble(row = rownames(m)[row(m)], 
                     col = colnames(m)[col(m)], corr = c(m)) %>%
  mutate(row = get_var_labs(row),
         col = get_var_labs(col))
heatmap_df %>%
  ggplot(aes(x = row, y = col, fill = corr)) + 
  geom_tile() + 
  scale_fill_continuous("Correlation") +
  theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust = 1),
        axis.title = element_blank()) 

6.4 Variable importance

Variable importance is assessed with a Wald test using the rms::anova().

# Compute variable important with Wald chi-square
lrm_anova_all <- anova(lrm_fit_all)

# Plot the result
lrm_anova_all %>% 
  as_tibble() %>%
  mutate(var = gsub(" ", "", rownames(lrm_anova_all)),
         varlab = get_var_labs(var),
         value = as.double(`Chi-Square` - `d.f.`)) %>%
  filter(!var %in% c("TOTAL", "Nonlinear", "TOTALNONLINEAR")) %>%

  ggplot(aes(x = value, y = reorder(varlab, value))) +
  geom_point() +
  theme(axis.title.y = element_blank()) +
  xlab(expression(chi^2-df)) 

6.5 Summary of odds ratios

The model is summarized with odds ratios using rms::summary(). We will start by assessing the full model. Next, since labs and vitals may themselves be “caused” by comorbidities, we will see how the odds ratios for the comorbidities change after dropping labs and vitals from the model.

6.5.1 Full model

lrm_summary_all <- summary(lrm_fit_all)

# Odds ratios
format_or_range <- function(x, term){
  case_when(
    x < 10 ~ formatC(x, format = "f", digits = 2),
    term == "temp" ~ formatC(x, format = "f", digits = 1),
    TRUE ~ formatC(x, format = "f", digits = 0)
  )
}

make_tidy_or <- function(object, model_name = NULL){
  if (is.null(model_name)) model_name <- "Model"
  object %>%
    as.data.frame() %>%
    as_tibble() %>%
    mutate(term = rownames(object),
           High = format_or_range(High, term),
           Low = format_or_range(Low, term),
           termlab = get_term_labs(term, "term2"),
           termlab = ifelse(!is.na(`Diff.`), 
                            paste0(termlab, " - ", High, ":", Low),
                            termlab),
           or = exp(Effect),
           or_lower = as.double(exp(`Lower 0.95`)),
           or_upper = exp(`Upper 0.95`)) %>%
    filter(Type == 1) %>%
    select(term, termlab, or, or_lower, or_upper) %>%
    arrange(-or) %>%
    mutate(model = model_name)
}
lrm_or_all <- make_tidy_or(lrm_summary_all, "All variables") 

# Odds ratio plot
ggplot(lrm_or_all, 
       aes(x = or, y = reorder(termlab, or))) +
  geom_point() +
  geom_errorbarh(aes(xmax = or_upper, xmin = or_lower,
                     height = .2)) +
  geom_vline(xintercept = 1, linetype = "dashed", col = "grey") + 
  theme(axis.title.y = element_blank()) +
  xlab("Odds ratio")

6.5.2 Comparison of model with and without vitals + labs

lrm_summary_dc <- summary(lrm_fit_dc)
lrm_or_dc <- make_tidy_or(lrm_summary_dc, "Demographics + comorbidities")

# Odds ratio comparison plot
lrm_or_comp <- bind_rows(lrm_or_all, lrm_or_dc) %>%
  filter(term %in% 
           terms[terms$group %in% c("Demographics", "Comorbidities"), ]$term2) %>%
  mutate(termlab = factor(termlab),
         termlab = reorder(termlab, or, function (x) -mean(x)))

ggplot(lrm_or_comp, 
       aes(x = termlab, y = or, col = model)) +
  geom_point(position = position_dodge(width = 1)) +
  geom_errorbar(aes(ymax = or_upper, ymin = or_lower,
                     width = .2), position = position_dodge(width = 1)) +
  facet_wrap(~termlab, strip.position = "left", ncol = 1, scales = "free_y") + 
  geom_hline(yintercept = 1, linetype = "dashed") + 
  theme(axis.title.y = element_blank()) +
  scale_color_discrete(name = "Model") +
  theme(axis.text.y = element_blank(),
        axis.ticks.y = element_blank(),
        strip.text.y.left = element_text(hjust = 0, vjust = 1, 
                                         angle = 0, size = 8),
        legend.position = "bottom") +
  ylab("Odds ratio") +
  coord_flip()

6.6 Predicted log odds

One limitation of the odds ratio plots is that it is hard to examine the potentially non-linear relationships between mortality and the continuous variable. The rms::Predict() function is especially helpul in this regard. We use it to vary all of the predictors and plot predicted log odds across the different values of each predictor.

Each prediction is made with all other variables at their “Adjust to” value as specified with datadist() above.

t(dd$limits) %>%
  as_tibble() %>%
  mutate(Variable = get_var_labs(colnames(dd$limits))) %>%
  relocate(Variable, .before = "Low:effect") %>%
  filter(!is.na(Variable)) %>%
  arrange(Variable) %>%
  kable() %>%
  kable_styling()

Variable	Low:effect	Adjust to	High:effect	Low:prediction	High:prediction	Low	High
Acute myocardial infarction	NA	No	NA	No	Yes	No	Yes
Age	49	62	75	18	89	18	89
AIDS/HIV	NA	No	NA	No	Yes	No	Yes
Aspartate aminotransferase (AST)	25	36	57	6	157	4	157
Body Mass Index (BMI)	25.5150	29.6900	35.1300	13.4700	77.8428	11.8400	149.5000
C-reactive protein (CRP)	29.7	73.0	133.0	0.2	458.0	0.0	458.0
Calendar time	38	47	64	1	91	0	91
Cancer	NA	No	NA	No	Yes	No	Yes
Cerebrovascular disease	NA	No	NA	No	Yes	No	Yes
Chronic pulmonary disease	NA	No	NA	No	Yes	No	Yes
Congestive heart failure	NA	No	NA	No	Yes	No	Yes
Creatinine	0.80	1.00	1.40	0.23	3.24	0.19	3.24
Dementia	NA	No	NA	No	Yes	No	Yes
Diabetes	NA	No	NA	No	Yes	No	Yes
Ethnicity	NA	Hispanic	NA	Hispanic	Not Hispanic	Hispanic	Not Hispanic
Ferritin	204.0	464.0	994.3	4.0	3648.0	3.0	3648.0
Geographic division	NA	Pacific	NA	Pacific	West North Central	Pacific	West North Central
Heart rate	77.50000	87.50000	98.00000	37.89579	164.10421	20.00000	203.00000
Hemiplegia or paraplegia	NA	No	NA	No	Yes	No	Yes
Hypertension	NA	No	NA	No	Yes	No	Yes
Lactate dehydrogenase (LDH)	229.00	307.00	423.00	65.00	1050.00	24.99	1050.00
Lymphocyte count	0.7	1.0	1.4	0.0	3.5	0.0	3.5
Metastatic cancer	NA	No	NA	No	Yes	No	Yes
Mild liver disease	NA	No	NA	No	Yes	No	Yes
Moderate/severe liver disease	NA	No	NA	No	Yes	No	Yes
Neutrophil count	3.31	4.88	7.09	0.00	18.29	0.00	18.29
Oxygen saturation	94.0	96.0	98.0	58.5	100.0	26.0	100.0
Peptic ulcer disease	NA	No	NA	No	Yes	No	Yes
Peripheral vascular disease	NA	No	NA	No	Yes	No	Yes
Platelet count (PLT)	158	203	261	7	569	1	569
Race	NA	African American	NA	African American	Caucasian	African American	Caucasian
Race/Ethnicity	NA	Non-Hispanic white	NA	Non-Hispanic white	Non-Hispanic black	Non-Hispanic white	Non-Hispanic black
Renal disease	NA	No	NA	No	Yes	No	Yes
Respiration rate	18	20	22	5	56	2	99
Rheumatoid disease	NA	No	NA	No	Yes	No	Yes
Sex	NA	Male	NA	Male	Female	Male	Female
Smoking	NA	Never	NA	Never	Previous	Never	Previous
Systolic blood pressure	115.00000	126.00000	139.00000	44.15459	223.00000	30.00000	266.00000
Temperature	36.7	37.0	37.4	16.0	41.2	16.0	42.2
Troponin I	0.010000	0.010000	0.045625	0.000000	0.170000	0.000000	0.170000
White blood cell count (WBC)	4.90	6.67	9.10	0.30	21.70	0.20	21.70

Let’s make the predictions.

lrm_log_odds <- Predict(lrm_fit_all, ref.zero = TRUE)

# Get plotting data
p_log_odds <- ggplot(lrm_log_odds, sepdiscrete = "list")

# Continuous plot
log_odds_limit <- max(ceiling(abs(p_log_odds$continuous$data$yhat)))
log_odds_breaks <- seq(-log_odds_limit, log_odds_limit, 1)
p_log_odds_continuous <- p_log_odds$continuous$data %>%
  as_tibble() %>%
  mutate(varlab = get_var_labs(.predictor.)) %>%
  ggplot(aes(x = .xx., y = yhat)) +
  facet_wrap(~varlab, scales = "free_x", ncol = 4) +
  geom_line() +
  geom_ribbon(aes(ymin = lower, ymax = upper), alpha = 0.3) +
  geom_hline(yintercept = 0, linetype = "dashed", color = "grey") +
  ylab("Log odds") +
  scale_y_continuous(breaks = log_odds_breaks,
                     limits = c(-log_odds_limit, log_odds_limit)) +
  theme(axis.title.x = element_blank(), 
        strip.text = element_text(size = 7))

# Discrete plot
log_odds_limit <- max(ceiling(abs(p_log_odds$discrete$data$yhat)))
log_odds_breaks <- seq(-log_odds_limit, log_odds_limit, 1)
p_log_odds_discrete <- p_log_odds$discrete$data %>%
  as_tibble() %>%
  mutate(varlab = get_var_labs(.predictor.)) %>%
  ggplot(aes(x = yhat, y = .xx.)) +
  facet_wrap(~varlab, scales = "free_y", ncol = 4) +
  geom_point(size = .9) +
  geom_errorbarh(aes(xmin = lower , xmax = upper, height = 0)) +
  geom_vline(xintercept = 0, linetype = "dashed", color = "grey") +
  xlab("Log odds") +
  scale_x_continuous(breaks = log_odds_breaks,
                     limits = c(-log_odds_limit, log_odds_limit)) +
  theme(axis.title.y = element_blank(),
        strip.text = element_text(size = 7))

# Combine plots
grid.arrange(p_log_odds_discrete, p_log_odds_continuous,
             heights = c(5, 5))

## Warning: Removed 6 rows containing missing values (geom_errorbarh).

6.7 Predicted probabilities

The predicted log odds plot is a nice way to summarize non-linear effects, but it’s not the ultimate quantity of interest. We really care about predicted probabilities. But to compute a predicted probability we need to set all covariates in the model to certain values. Rather that choosing specific values (i.e., creating a representative patient), we will make predictions over a representative sample and average predictions across the sample.

# Make newdata
## Start with a random sample of patients
n_samples <- 1000
train_sample <- mi_list[[1]] %>% 
  filter(.imp > 0) %>%
  sample_n(size = n_samples)

We will write a general function to predict mortality as a function of (i) our fitted model, (ii) the representative sample, and (iii) the variables that we are varying.

expand_newdata <- function(data, vars_to_vary){
  varnames_to_vary <- names(vars_to_vary)
  expanded_vars_to_vary <- expand.grid(vars_to_vary)
  data %>% 
    select(-all_of(varnames_to_vary)) %>%
    crossing(expanded_vars_to_vary)
}

predict_mortality <- function(fit, newdata, vars_to_vary){
  expanded_newdata <- expand_newdata(newdata, vars_to_vary)
  fit_probs <- predict(fit, newdata = data.frame(expanded_newdata), 
                       type = "fitted", se.fit = FALSE)
  
  # Average predictions for variables to vary
  expanded_newdata %>%
    mutate(prob = fit_probs) %>%
    group_by(across(all_of(names(vars_to_vary)))) %>%
    summarise(prob = mean(prob))
}

predict_mortality_boot <- function(fit, newdata, vars_to_vary, B = 100){
  n_obs <- nrow(newdata)
  boot_probs <- vector(mode = "list", length = B)
  for (b in 1:B){
    indices <- sample(x = 1:n_obs, size = n_obs, replace = TRUE) 
    newdata_boot <- newdata[indices, ]
    boot_probs[[b]] <- predict_mortality(fit, newdata_boot, vars_to_vary)
    boot_probs[[b]][, "b"] <- b 
  }
  return(boot_probs %>% bind_rows())
}

6.7.1 Varying age

ages_to_vary  <- seq(min(train_data$age),  max(train_data$age), 1)

probs_age <- predict_mortality(lrm_fit_all, newdata = train_sample, 
                               vars_to_vary = list(age = ages_to_vary)) %>%
  rename(Age = age, `Mortality probability` = prob)

## `summarise()` ungrouping output (override with `.groups` argument)

# Table using DT package
datatable(probs_age, 
          rownames = FALSE,
          options = list(pageLength = 20)) %>%
  formatRound("Mortality probability", 4)

ggplot(probs_age, aes(x = `Age`, y = `Mortality probability`)) +
  geom_line() 

6.7.2 Varying calendar time

min_index_date <- min(train_data$index_date)
calendar_times_to_vary <- c(as.Date("2020-03-01"), 
                            as.Date("2020-04-01"), 
                            as.Date("2020-05-01")) - min_index_date 
calendar_times_to_vary <- as.numeric(calendar_times_to_vary)

probs_calendar_time <- predict_mortality(
  lrm_fit_all, newdata = train_sample, 
  vars_to_vary = list(calendar_time = calendar_times_to_vary)
) %>%
  mutate(Date = min_index_date + calendar_time) %>%
  rename(`Calendar time` = calendar_time, `Mortality probability` = prob) %>%
  relocate(Date)

## `summarise()` ungrouping output (override with `.groups` argument)

kable(probs_calendar_time) %>% kable_styling()

Date	Calendar time	Mortality probability
2020-03-01	10	0.2240193
2020-04-01	41	0.1710479
2020-05-01	71	0.1388932

6.7.3 Varying age, sex, and calendar time

We will start by predicting point estimates only.

sex_to_vary <- unique(train_data$sex) %>%
  .[!is.na(.)]

# Predict point estimates
probs_age_sex_calendar <- predict_mortality(
  lrm_fit_all, newdata = train_sample, 
  vars_to_vary = list(age = ages_to_vary,
                      sex = sex_to_vary,
                      calendar_time = calendar_times_to_vary))

## `summarise()` regrouping output by 'age', 'sex' (override with `.groups` argument)

# Summarize in table
probs_age_sex_calendar %>%
  arrange(calendar_time, sex, age) %>%
  mutate(Date = min_index_date + calendar_time) %>%
  select(-calendar_time) %>%
  relocate(Date, sex) %>%
  rename(`Sex` = sex, `Age` = age,
         `Mortality probability` = prob) %>%
  
  ## Using DT::datatable
  datatable(rownames = FALSE,
            filter = "top",
          options = list(pageLength = 20)) %>%
  formatRound("Mortality probability", 4)

We we will also compute 95% confidence intervals by bootstrapping.

bootprobs_age_sex_calendar <- predict_mortality_boot(
  lrm_fit_all, newdata = train_sample, 
  vars_to_vary = list(age = ages_to_vary,
                      sex = sex_to_vary,
                      calendar_time = calendar_times_to_vary),
  B = n_boot_probs) 

Let’s plot the results.

bootprobs_age_sex_calendar %>%
  # Summarize bootstrap results
  group_by(age, sex, calendar_time) %>%
  summarise(
    prob_mean = mean(prob),
    prob_lower = quantile(prob, .025),
    prob_upper = quantile(prob, .975)
  ) %>%
  mutate(date = min_index_date + calendar_time) %>%

  # Plot
  ggplot(aes(x = age, y = prob_mean, color = sex, fill = sex)) +
    geom_line() +
    facet_wrap(~date) +
    geom_line(aes(x = age, y = prob_lower, color = sex),  linetype = "dotted") +
    geom_line(aes(x = age, y = prob_upper, color = sex),  linetype = "dotted") +
    xlab("Age") +
    ylab("Mortality probability") +
    scale_color_discrete("") +
    scale_fill_discrete("") +
    theme(legend.position = "bottom")

## `summarise()` regrouping output by 'age', 'sex' (override with `.groups` argument)

7 Model validation

7.1 Optimism-adjusted model performance

We will start by using bootstrapping to estimate model performance and check for whether our in-sample fits are too optimistic. Specifically, we will use the following algorithm implemented in the rms package:

1. Calculate performance from the model fit on the original training sample, denoted as \(\theta_{orig}\).
2. For \(b = 1,\ldots,B\):
   1. Bootstrap (with replacement) from the original training sample.
   2. Fit the model to the bootstrapped data and estimate model performance, denoted as \(\theta_{train}\).
   3. Use the bootstrap fit to measure model performance on the original training sample, denoted as \(\theta_{test}\).
3. Calculate “optimism” as \(O = B^{-1} \sum_{b = 1}^B \theta_{b, train} - \theta_{b, test}\).
4. Calculate optimism-adjusted performance as \(\theta = \theta_{orig} - O\).

A shrinkage factor can also be estimated within each bootstrap sample to gauge the extent of overfitting. This is done by fitting \(g(Y) = \gamma_0 + \gamma_1 X\hat{\beta}\) where \(X\) and \(Y\) are the covariates and outcome, respectively, in the test sample (i.e., in step 2c) and \(\hat{\beta}\) is estimating in the training sample (i.e., step 2b). If there is no overfitting, then \(\gamma_0 = 0\) and \(\gamma_1 = 1\); conversely, if there is overfitting, then \(\gamma_1 < 1\) and \(\gamma_0 \neq 1\) to compensate.

lrm_val_age <- validate(lrm_fit_age, B = n_boot_val)
lrm_val_c <- validate(lrm_fit_c, B = n_boot_val) 
lrm_val_d <- validate(lrm_fit_d, B = n_boot_val)
lrm_val_dc <- validate(lrm_fit_dc, B = n_boot_val)
lrm_val_all <- validate(lrm_fit_all, B = n_boot_val) 

bind_cindex <- function(object){
  n_rows <- nrow(object)
  c_index <- (object["Dxy", 1:3] + 1)/2
  c_index[4] <- c_index[2] - c_index[3]
  c_index[5] <- c_index[1] - c_index[4]
  c_index[6] <- object[1, 6]
  
  return(rbind(object, c_index))
}

make_validation_tbl <- function(object){
  object %>% 
    bind_cindex() %>%
    set_colnames(c("(1) Original", "(2) Bootstrap training", 
                   "(3) Bootstrap test", "Optimism: (2) - (3)", 
                   "Original (corrected): (1) - (4)", "N")) %>%
    kable() %>%
    kable_styling()
}

make_validation_tbl(lrm_val_age)

	Original	Bootstrap training	Bootstrap test	Optimism: (2) - (3)	Original (corrected): (1) - (4)	N
Dxy	0.5491275	0.5474620	0.5491275	-0.0016655	0.5507929	40
R2	0.2120752	0.2109979	0.2120752	-0.0010773	0.2131525	40
Intercept	0.0000000	0.0000000	0.0060778	-0.0060778	0.0060778	40
Slope	1.0000000	1.0000000	1.0049457	-0.0049457	1.0049457	40
Emax	0.0000000	0.0000000	0.0021623	0.0021623	0.0021623	40
D	0.1318283	0.1310950	0.1318283	-0.0007333	0.1325616	40
U	-0.0001464	-0.0001464	0.0000326	-0.0001790	0.0000326	40
Q	0.1319747	0.1312414	0.1317957	-0.0005543	0.1325290	40
B	0.1160266	0.1160576	0.1160503	0.0000073	0.1160193	40
g	1.3960983	1.3911119	1.3960983	-0.0049864	1.4010848	40
gp	0.1458584	0.1453858	0.1458584	-0.0004726	0.1463310	40
c_index	0.7745637	0.7737310	0.7745637	-0.0008327	0.7753965	40

make_validation_tbl(lrm_val_c) 

	Original	Bootstrap training	Bootstrap test	Optimism: (2) - (3)	Original (corrected): (1) - (4)	N
Dxy	0.4620296	0.4629949	0.4555613	0.0074336	0.4545960	40
R2	0.1382186	0.1409290	0.1365210	0.0044080	0.1338106	40
Intercept	0.0000000	0.0000000	-0.0201600	0.0201600	-0.0201600	40
Slope	1.0000000	1.0000000	0.9822041	0.0177959	0.9822041	40
Emax	0.0000000	0.0000000	0.0075298	0.0075298	0.0075298	40
D	0.0838930	0.0854538	0.0828178	0.0026360	0.0812570	40
U	-0.0001464	-0.0001464	0.0000386	-0.0001850	0.0000386	40
Q	0.0840394	0.0856002	0.0827793	0.0028210	0.0812184	40
B	0.1215780	0.1208416	0.1217908	-0.0009491	0.1225271	40
g	0.7727543	0.7823849	0.7677599	0.0146250	0.7581293	40
gp	0.1097598	0.1104412	0.1090014	0.0014398	0.1083200	40
c_index	0.7310148	0.7314975	0.7277806	0.0037168	0.7272980	40

make_validation_tbl(lrm_val_all)

	Original	Bootstrap training	Bootstrap test	Optimism: (2) - (3)	Original (corrected): (1) - (4)	N
Dxy	0.7759059	0.7792427	0.7719964	0.0072463	0.7686596	40
R2	0.4542152	0.4560254	0.4454843	0.0105411	0.4436741	40
Intercept	0.0000000	0.0000000	-0.0253335	0.0253335	-0.0253335	40
Slope	1.0000000	1.0000000	0.9730032	0.0269968	0.9730032	40
Emax	0.0000000	0.0000000	0.0104527	0.0104527	0.0104527	40
D	0.3073570	0.3090945	0.3004598	0.0086347	0.2987223	40
U	-0.0001464	-0.0001464	0.0001235	-0.0002699	0.0001235	40
Q	0.3075034	0.3092409	0.3003363	0.0089046	0.2985988	40
B	0.0880580	0.0874447	0.0887493	-0.0013046	0.0893626	40
g	2.3829653	2.4116410	2.3446519	0.0669891	2.3159762	40
gp	0.2048515	0.2055457	0.2030382	0.0025075	0.2023441	40
c_index	0.8879530	0.8896214	0.8859982	0.0036232	0.8843298	40

summarize_performance <- function(objects){
  metrics <- c("Dxy", "R2", "B")
  map_df(objects, function (x) x[metrics, "index.orig"],
                .id = "Model") %>%
    mutate(`C-index` = (Dxy + 1)/2) %>%
    rename(`Brier score` = B, `R-Squared` = R2) %>%
    select(-Dxy) %>%
    kable(digits = 3) %>%
    kable_styling()
}

list(lrm_val_age, lrm_val_c, lrm_val_d, lrm_val_dc, lrm_val_all) %>%
  set_names(lrm_names) %>%
  summarize_performance() 

Model	R-Squared	Brier score	C-index
Age only	0.212	0.116	0.775
Comorbidities only	0.138	0.122	0.731
All demographics	0.229	0.114	0.784
Demographics and comorbidities	0.256	0.112	0.802
All variables	0.454	0.088	0.888

7.2 Calibration curves

lrm_cal_age <- calibrate(lrm_fit_age, B = n_boot_val)
lrm_cal_c <- calibrate(lrm_fit_c, B = n_boot_val)
lrm_cal_d <- calibrate(lrm_fit_d, B = n_boot_val)
lrm_cal_dc <- calibrate(lrm_fit_dc, B = n_boot_val)
lrm_cal_all <- calibrate(lrm_fit_all, B = n_boot_val)
lrm_cal_list <- list(lrm_cal_age, lrm_cal_c, lrm_cal_d, lrm_cal_dc, lrm_cal_all)
names(lrm_cal_list) <- lrm_names

plot_calibration <- function(object){
  # Make tibble
  cal_df <- map2(object, names(object), function(x, y){
    x[, ] %>%
      as_tibble() %>%
      mutate(model = y)
  }) %>% 
    bind_rows() %>%
    mutate(model = factor(model, levels = lrm_names))
  
  
  # Plot
  breaks <- seq(0, 1, .2)

  ggplot() + 
    geom_line(data = cal_df, mapping = aes(x = predy, y = calibrated.orig, 
                                           color = "Apparent")) + 
    geom_line(data = cal_df, mapping = aes(x = predy, y = calibrated.corrected,
                                           color = "Bias-corrected")) +
    geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "grey") +
    facet_wrap(~model) +
    scale_x_continuous(breaks = breaks, limits = c(0, 1)) +
    scale_y_continuous(breaks = breaks, limits = c(0, 1)) +
    xlab("Predicted probability") +
    ylab("Actual probability") +
    scale_colour_manual(name = "",
                        values = c("Apparent" = "black", 
                                   "Bias-corrected" = "red")) +
    theme(legend.position = "bottom") 
}

plot_calibration(lrm_cal_list)

7.3 Performance on test set